Controlling Several Atoms in a Cavity

arXiv:1401.5722v1 [quant-ph] 22 Jan 2014

Controlling Several Atoms in a Cavity Michael Keyl Zentrum Mathematik, M5, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany E-mail: [email protected]

Robert Zeier Department Chemie, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching, Germany E-mail: [email protected]

Thomas Schulte-Herbrüggen Department Chemie, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching, Germany E-mail: [email protected] (Dated: January 22, 2014) PACS numbers: 03.67.Ac, 02.30.Yy, 42.50.Ct, 03.65.Db Abstract. We treat control of several two-level atoms interacting with one mode of the electromagnetic field in a cavity. This provides a useful model to study pertinent aspects of quantum control in infinite dimensions via the emergence of infinite-dimensional system algebras. Hence we address problems arising with infinitedimensional Lie algebras and those of unbounded operators. For the models considered, these problems can be solved by splitting the set of control Hamiltonians into two subsets: The first obeys an abelian symmetry and can be treated in terms of infinite-dimensional Lie algebras and strongly closed subgroups of the unitary group of the system Hilbert space. The second breaks this symmetry, and its discussion introduces new arguments. Yet, full controllability can be achieved in a strong sense: e.g., in a time dependent Jaynes-Cummings model we show that, by tuning coupling constants appropriately, every unitary of the coupled system (atoms and cavity) can be approximated with arbitrarily small error.

Controlling Several Atoms in a Cavity

2

Contents 1 Introduction 2 Controllability 2.1 Time evolution . . . . . 2.2 Pure-state controllability 2.3 Strong controllability . . 2.4 The dynamical group G 2.5 Abelian symmetries . . 2.6 Breaking the symmetry

2

. . . . . .

3 3 4 5 5 6 8

3 Atoms in a cavity 3.1 One atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Many atoms with individual control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Many atoms under collective control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 11 13

4 A Lie algebra of block-diagonal operators 4.1 Commuting operators . . . . . . . . . . . 4.2 One atom . . . . . . . . . . . . . . . . . . 4.3 Many atoms with individual control . . . 4.4 Many atoms under collective control . . .

. . . .

14 15 19 21 23

5 Strong controllability 5.1 Pure-state controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Approximating unitaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Strong controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26 26 28 29

6 Conclusions and Outlook

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1. Introduction Exploiting controlled dynamics of quantum systems is becoming of increasing importance not only for solving computational tasks or quantum-secured communication, but also for simulating other physical systems [1, 2, 3, 4, 5, 6]. An interesting direction in quantum simulation applies many-body correlations to create ‘quantum matter’. E.g., ultra-cold atoms in optical lattices are versatile models for studying large-scale correlations [7, 6]. Tunability and control over the system parameters of optical lattices allows for switching between several low-energy states of different quantum phases [8, 9] or in particular for following real-time dynamics such as the quantum quench from the super-fluid to the Mott insulator regime [10]. Thus manipulating several atoms in a cavity is a key step to this end [11] at the same time posing challenging infinite-dimensional control problems. While in finite dimensions controllability can readily be assessed by the Lie-algebra rank condition [12, 13, 14, 15, 16], infinite-dimensional systems are more intricate [17]. As exact controllability in infinite dimensions seemed daunting in earlier work [18, 19, 20, 21], it took a while before approximate control paved the way to more realistic assessment [22, 23, 24], for a recent (partial) review see, e.g., also [25] and references therein. Here we explore systems and control aspects for systems consisting of several twolevel atoms coupled to a cavity mode, i.e. the Jaynes-Cummings model [26, 27, 28, 29]. We build upon our previous symmetry arguments [30, 31] and moreover, we apply appropriate operator topologies for addressing two controllability problems in particular:


3

(i) to which extent can pure states be interconverted and (ii) can unitary gates be approximated with arbitrary precision. In particular by treating the latter, we go beyond previous work, which started out by a finite-dimensional truncation of a two-level atom coupled to an oscillator [32] followed by generalisations to infinite dimensions [33, 34, 35] both being confined to establishing criteria of pure-state controllability. Note that [35] also treats one atom coupled to several oscillators. The general aim of this paper is twofold: On the one hand we study control problems for atoms interacting with electromagnetic fields in cavities. On the other hand, we address quantum control in infinite dimensions. Therefore, the purpose of Section 2 is to provide enough material for a non-technical overview on the second subject in order to understand the results on the first (where the difficulties come from). Mathematical details are postponed to Sections 4 and 5, while results on cavity systems are presented in overview in Section 3. 2. Controllability The control of quantum systems poses considerable mathematical challenges when applied to infinite dimensions. Basically, they arise from the fact that anti-selfadjoint operators (recall that according to Stone’s Theorem [36, VIII.4], they are generators of strongly continuous, unitary one-parameter groups) do neither form a Lie algebra nor even a vector space. Or seen on the group level, the group of unitaries equipped with the strong operator topology is a topological group yet not a Lie group. So whenever strong topology has to be invoked, controllability cannot be assessed via a system Lie algebra. Thus we address these challenges on the group level by employing the controlled time evolution of the quantum system in order to approximate unitary operators, the action of which is measured with respect to arbitrary, but finite sets of vectors. This is formalized in the notion of strong controllability (see Section 2.3) introduced here as a generalisation of pure-state controllability already discussed in the literature. Central to our discussion are abelian symmetries. Assuming that all but one of our Hamiltonians observe such an ablian symmetry, we systematically analyze the infinite-dimensional control system in its block-diagonalized basis. We obtain strong controllability (beyond pure-state controllability) if one of the Hamiltonian breaks this abelian symmetry and some further technical conditions are fullfilled. 2.1. Time evolution We treat control problems of the form X ˙ = ψ(t) uk (t)Hk ψ(t) = H(t)ψ(t)

(1)

k

where the Hk with k ∈ {1, . . . , d} are selfadjoint control Hamiltonians on an infinitedimensional, separable Hilbert space H and the controls uk : R → R are piecewiseconstant control functions. Since H is infinite-dimensional, the operators Hk are usually


4

only defined on a dense subspace D(Hk ) ⊂ H called the domain of Hk , the only exceptions being those Hk which are bounded. However, in this context, control problems where all Hk are bounded are not very interesting from a physical point of view. In other words, there is no way around considering those domains and many difficulties of control theory in infinite dimensions arises from this fact‡ . We will also assume that Eq. (1) will have unique solutions for all initial states ψ0 ∈ H and all times t. So for each pair of times t1 < t2 there is a unitary propagator Rt U(t1 , t2 )ψ0 = T t12 exp(−itH(t))ψ0 , where T denotes time ordering. Observe that this condition is usually not satisfied, not even if the Hk share a joint domain of essential selfadjointness. Fortunately, the systems we are going to study do not show such pathological behavior. Yet, a minimalistic way to avoid this problem would be to restrict to control functions where only one uk is different from 0 at each time t. In this case the propagator U(t1 , t2 ) is just a concatenation of unitaries exp(itHk ) which are guaranteed to exist due to selfadjointness of the Hk . 2.2. Pure-state controllability A key-issue in quantum control theory is reachability: Given two pure states ψ0 , ψ ∈ H, we are looking for a time T > 0 and control functions uk such that ψ = U(0, T )ψ0 . In infinite dimensions, however, this condition is too strong, since there might be states which can be reached only in infinite time, or not at all. Yet, one may find a reachable state “close by” with arbitrary small control error. Therefore we will call ψ reachable from ψ0 if for all ǫ > 0 there is a finite time T > 0 and control functions uk such that kψ − U(0, T )ψ0 k < ǫ holds. Accordingly, we will call the system (1) pure-state controllable, if each pure state ψ can be reached from one ψ0 (and, by unitarity, also vice versa). Since pure states are described by one-dimensional projections, two state vectors describe the same state if they differ only by a global phase. Hence the definition just given is actually a bit too strong. There are several ways around this problem, like using the trace norm distance of |ψihψ| and |ψ0 ihψ0 | rather then the norm distance of ψ and ψ0 . For our purposes, however, the most appropriate method is to assume that the unit operator 1 on H is always among the control Hamiltonians. This may appear somewhat arbitrary, but it helps to avoid problems with determinants and traces on infinite-dimensional Hilbert spaces, which otherwise would arise. ‡ Note that domains of unbounded operators are not just a mathematical pedantism. The domain is a crucial part of the definition of an operator and contains physically relevant information. A typical example is the Laplacian in a box which requires boundary conditions for a complete description. Up to a certain degree, domains can be regarded as an abstract form of boundary conditions (possibly at infinity).


5

2.3. Strong controllability Next, the analysis shall be lifted to the level of operators, i.e. to unitaries U from the group U(H) of unitary operators on the Hilbert space H such that a time T > 0 and control functions uk exist with U = U(0, T ). As in the last paragraph, this has to be generalized to an approximative condition again. The best choice—mathematically as well as from a practical point of view—is approximation in the strong sense: We look for unitaries U such that for each set of (not necessarily orthonormal or linearly independent) vectors ψ1 , . . . , ψf ∈ H and each ǫ > 0, there exists a time T > 0 and control functions uk such that k[U − U(0, T )] ψk k < ǫ for all k ∈ {1, . . . , f }.

(2)

In other words, we are comparing U and U(0, T ) only on a finite set of states, and the worst-case error one can get here is bounded by ǫ. We will call the control system (1) strongly controllable if each unitary U can be approximated that way. (NB, in strong controllability, one again has the choice of one single joint global phase factor.) Clearly, strong controllability implies pure-state controllability. To see this, choose an arbitrary but fixed ψ0 ∈ H. For each ψ ∈ H, there is a unitary U with Uψ0 = ψ. Hence strong controllability implies kψ − U(0, T ) ψ0 k = k[U − U(0, T )] ψ0 k < ǫ. 2.4. The dynamical group G Strong controllability is concept-wise related to the strong operator topology [36, VI.1] on the group U(H) of unitary operators on H. To this end, consider the sets N (U; ψ1 , . . . , ψf ; ǫ) = {V ∈ U(H) | k(V − U) ψk k < ǫ for all k ∈ {1, . . . , f }}.

(3)

They form a neighborhood base for the strong topology, and we will call them (strong) ǫ-neighborhoods. The condition in Eq. (2) can now be restated as: Any ǫ-neighborhood of U contains a time-evolution operator U(0, T ) for appropriate time T and control functions uk . In turn, this can be reformulated as: U is an accumulation point of the set G˜ of all unitaries U(0, T ). The set of all accumulation points of G˜ (which contains G˜ itself) is a strongly closed subgroup§ of U(H), which we will call the dynamical group G generated by control Hamiltonians Hk with k ∈ {1, . . . , d}. If we choose the controls as described in Subsection 2.1 (i.e. piecewise constant and only one uk different from zero at each time), G is just the smallest strongly closed subgroup of U(H) that contains all exp(itHk ) for all k ∈ {1, . . . , d} and all t ∈ R. Note that it contains in particular all unitaries that can be written as a strong limit s-limT →∞ U(0, T ). In finite dimensions, § There is a subtle point here: The group U(H) is not strongly closed as a subset of the bounded operators B(H). Actually its strong closure is the set of all isometries; cf. [37, Prob. 225]. Hence whenever we talk about strongly closed groups of unitaries, this has to be understood as the closure in the restriction of the strong topology to U(H) (which coincides with the restriction of the weak topology).


6

G can be calculated via its system algebra, i.e. the Lie algebra l generated by the iHk , since each U ∈ G can be written as U = exp(H) for an H ∈ l. In infinite dimensions, however, several difficulties can occur. First, unbounded operators Hk are only defined on a dense domain D(Hk ) ⊂ H. The sum Hk + Hj is therefore only defined on the intersection D(Hk )∩D(Hj ) and the commutator even only on a subspace thereof. There is no guarantee that D(Hk ) ∩ D(Hj ) contains more than just the zero vector. In this case, the Lie algebra cannot even be defined. The minimal requirement to get around this difficulty is the existence of a joint dense domain D, i.e. D ⊂ D(Hj ) and Hj D ⊂ D for all j. However, even then we do not know whether G can be generated from l in terms of exponentials. In general, it is impossible to define some exp(H) for all H ∈ l. There are several ways to deal with these problems. One is to consider cases where the Hk generate (i) a finite-dimensional Lie algebra and admit (ii) a common, invariant, dense domain consisting of analytic vectors [18, 20]. In this case the exponential function is defined on all of l, and we can proceed in analogy to the finite-dimensional case. The problem is that the group G will become a finite-dimensional Lie group and its orbits through a vector ψ ∈ H are finite-dimensional as well. Hence, we never can achieve full controllability. This approach is well studied; cf. [18, 20] and references therein. Another possibility which includes the possibility to study an infinite-dimensional Lie algebra l is to restrict to bounded generators Hk . In this case, one can define l as a norm-closed subalgebra of the Lie algebra B(H) of bounded operators, and one ends up with a Banach-space theory which works almost in the same way as the finitedimensional analog; cf. [38] for details. Although this is a perfectly reasonable approach from the mathematical point of view, it is not very useful for physical applications, since in most cases at least some of the Hk are unbounded. In this paper, we will thus consider a different approach which splits the generators into two classes. The first d−1 generators H1 , . . . , Hd−1 admit an abelian symmetry and can be treated—with Lie-algebra methods—along the lines outlined in the next subsection. Secondly, the last generator Hd breaks this symmetry and achieves full controllability with a comparably simple argument. The details will be explained in Section 4 and 5. 2.5. Abelian symmetries One way to avoid the problem described in the last subsection, arises if the control system admits symmetries. In this section, we will only sketch the structure, while the details are postponed to Sect. 4. Let us consider the case of a U(1)-symmetryk, i.e. a (strongly continuous) unitary representation z 7→ π(z) ∈ U(H) of the abelian group U(1) on H where U(H) denotes the group of unitaries on H. It can be written in terms of a selfadjoint operator X with pure point spectrum consisting of (a subset of) Z as U(1) ∋ z = eiα 7→ π(z) = k The generalization to multiple charges, i.e. a U(1)N , is straightforward.


7

exp(iαX) ∈ U(H). If we denote the eigenprojection of X belonging to the eigenvalue µ ∈ Z as X (µ) (allowing the case X (µ) = 0 if µ is not an eigenvalue of X) we get a block-diagonal decomposition of H in the symmetry-adapted basis as H=

∞ M µ=−∞

H(µ) with H(µ) = X (µ) H,

(4)

P iαµ (µ) X ∈ U(H). and we can rewrite π(z) again as U(1) ∋ z = eiα 7→ π(z) = ∞ µ=−∞ e Here we will make two assumptions representing substantial restrictions of generality: (i) All eigenvalues of X are of finite multiplicity, i.e. the H(µ) are finite-dimensional. This is crucial for basically everything we will discuss in this paper. (ii) All eigenvalues of X are non-negative. This assumption can be relaxed at certain points (e.g. all material in Sect. 4.1 can be easily generalized). However, it helps to simplify the discussion at a technical level and all examples we are going to consider in the next section are of this form. The first important consequence of (i) concerns the space of finite particle vectors DX = {ψ ∈ H | X (µ) ψ = 0 for all but finitely many µ},

(5)

since it becomes (due to finite-dimensionality of H(µ) ) a “good” domain for basically all unbounded operators appearing in this paper. Moreover one gets the following theorem: Theorem 2.1. Consider a strongly continuous representation π of U(1) on H and the corresponding charge-type operator X. Then the following statements hold: (i) A selfadjoint operator H commuting with X admits DX as an invariant domain, i.e. DX ⊂ D(H) and HDX = DX . Hence the space u(X) = {iH | H = H ∗ commuting with X} is a Lie algebra with the commutator as its Lie bracket. (ii) The exponential map is well defined on u(X) and maps it onto the strongly closed subgroup U(X) = {U ∈ U(H) | [U, π(z)] = 0 for all z ∈ U(1)} of U(H).

(iii) The subalgebra l ⊂ u(X) generated by a family of Hamiltonians iH1 , . . . , iHd ∈ u(X) is mapped by the exponential map into the dynamical group G of the corresponding control problem. The strong closure of exp(l) coincides with G. The basic idea behind this theorem, is that one can cut off the decomposition (4) at a sufficiently high µ without sacrificing strong approximations as described in Subsection 2.3. One only has to take into account that the cut-off on µ has to become higher when the approximation error decreases. This strategy allows for tracing a lot of calculations back to finite-dimensional Lie algebras. We will postpone a detailed discussion of this topic—including the proof of Theorem 2.1—to Section 4. The only additional material one needs at this point, since it is of relevance for the next section, is a subgroup of U(X) and its corresponding Lie algebra which relates unitaries with determinant one and their traceless generators. Since the iH ∈ u(X) are unbounded and not necessarily positive, it is difficult to give a reasonable definition of


8

tracelessness, and the determinant of U ∈ U(X) runs into similar problems. However, the elements of U ∈ U(X) and iH ∈ u(X) are block diagonal with respect to the P P decomposition of H given in (4). In other words U = µ U (µ) and H = µ H (µ) are infinite sums of operators¶, where U (µ) = X (µ) UX (µ) ∈ U(H(µ) ), H (µ) = X (µ) HX (µ) ∈ B(H(µ) ), and X (µ) denotes the projection onto the X-eigenspace H(µ) . Since all the U (µ) and H (µ) are operators on finite-dimensional vector spaces, one can define SU (X) := {U ∈ U(X) | det U (µ) = 1 for all µ ∈ Z},

su(X) := {iH ∈ u(X) | tr(H (µ) ) = 0 for all µ ∈ Z}.

(6) (7)

Obviously, SU (X) is a (strongly closed) subgroup of U(X) and su(X) is a Lie subalgebra of u(X). The image of su(X) under the exponential map therefore coincides with SU (X). Note that SU (X) is effectively an infinite direct product of groups SU(d(µ) ), if d(µ) = dim H(µ) and not the “special” subgroup of U(X). 2.6. Breaking the symmetry To get a fully controllable system, one has to leave the group U(X), which can be thought of as being represented as block diagonal, see Fig. 1 a. To this end, we have to add control Hamiltonians that break the symmetry. There are several ways of doing so, and a successful strategy depends on the system in question (beyond the treatment of the symmetric part of the dynamics captured in Theorem 2.1). Here, we will present a special result which covers the examples discussed in the next section. The first step is another direct sum decomposition of H = H− ⊕ H0 ⊕ H+ , where Hα = Eα H, with α ∈ {+, 0, −} are projections onto the subspaces Hα and should satisfy [Eα , X (µ) ] = 0. Let in the following N := {1, 2, 3, . . .} denote the set of positive integers and define (µ) N0 := N ∪ {0}. Hence for µ ∈ N0 we can introduce the projections X± = X (µ) E± (0) which we require to be non-zero. For the exceptional case µ = 0 the relation X− = (µ) X (0) E− = X (0) should hold. Futhermore we write X0 = X (µ) E0 for the overlap of (µ) (µ) X (µ) and E0 which can (in contrast to X± ) be equal to zero for all µ. The Xα are (µ) (µ) (µ) (µ) (µ) projections onto the subspaces Hα := Xα H satisfying X (µ) = X− ⊕ X0 ⊕ X+ . Definition 2.2. A selfadjoint operator H with domain D(H) is called complementary to X, if there exists a decompositon H = H− ⊕ H0 ⊕ H+ as defined above such that: (i) H0 ⊂ D(X) and Hψ = 0 for all ψ ∈ H0 .

(µ+1)

(µ)

(ii) DX ⊂ D(H) and for all µ > 1 we have HX+ ψ = X− Hψ. The corresponding (µ) (µ+1) (µ+1) operator X− HX+ ∈ B(H) is a partial isometry with X+ as its source and (µ) X− as its target projection. ¶ Two small remarks are in order here: (i). Infinite sums require a proper definition of convergence in an appropriate topology. In Section 4, this will be made precise. (ii). Operator products of the form X (µ) HX (µ) are potentially problematic if H is unbounded and therefore only defined on a domain. In our case, however, X (µ) projects onto H(µ) , which is a subspace of the domain DX on which H is defined.

Controlling Several Atoms in a Cavity (a)

9 (b)

block structure of operators commuting with X

|↑↑i

(1)

ωI ωA

(2)

ωI |↓↑i |↑↓i

(2)

ωI

ωA block structure of operators complementary to X

|↓↓i

|n = 0i

(1)

ωI

|n = 1i

|n = 2i

Figure 1. (a) Block structure of operators in u(X) (red) and of operators complementary to X (blue) in the case where the projection E0 vanishes. (b) Energy diagram for the Jaynes-Cummings model (here two atoms in a cavity (2) (1) under individual controls ωI and ωI ) with combined atom-cavity transitions matching the block structure of (a) given in red (see Eqs. (10,18)) since commuting with X1 or XM of Eqs. (12, 20), and complementary transitions solely within the atoms given in blue (see Eqs. (15,22)). (0)

(1)

(iii) Given the projection F[0] = X ⊕X− and the corresponding subspace H[0] = F[0] H. The group generated by exp(itH) with t ∈ R and those U ∈ SU (X) which commute with F[0] acts transitively on the space of one-dimensional projections in H[0] . At first sight, the definition may look somewhat clumsy, but it allows for proving a controllability result which covers all examples we are going to present in the next section. We will state them here without a proof and postpone the latter to Sect. 5. Theorem 2.3. Consider a strongly continuous representation π : U(1) → U(H) with charge operator X and a family of selfadjoint operators H1 , . . . , Hd on H. Assume that the following conditions hold: (i) H1 , . . . , Hd−1 commute with X. (ii) The dynamical group generated by H1 , . . . , Hd−1 contains SU (X).

(iii) The operator Hd is complementary to X.

Then the control system (1) with Hamiltonians H0 = 1, H1 , . . . , Hd is pure-state controllable. Theorem 2.4. The control system (1) is even strongly controllable if in addition to the assumptions of Thm. 2.3 the condition dim H(µ) > 2 holds for at least one µ ∈ N0 . 3. Atoms in a cavity An important class of examples that can be treated along the lines described in the last section are atoms interacting with the light field in a cavity. We will discuss the case of M two-level atoms interacting with one mode in detail and consider three particular


10

scenarios: one atom in Sect. 3.1, individually controlled atoms in Sect. 3.2, and atoms under collective control in Sect. 3.3. 3.1. One atom Let us start with the special case M = 1, i.e. one atom and one mode as discussed in a number of previous publications mostly on pure-state controllability [39, 35, 34]. Our results go beyond this, in particular because we are considering strong controllability not just pure-state controllability. The Hilbert space of the system is given by H = C2 ⊗ L2 (R)

(8)

and the dynamics is described by the well known Jaynes-Cummings Hamiltonian [26]: HJC := ωA HJC,1 + ωI HJC,2 + ωC HJC,3 with

(9)

HJC,1 := (σ3 ⊗ 1)/2, HJC,2 := (σ+ ⊗ a + σ− ⊗ a∗ )/2, HJC,3 := 1 ⊗ N,

(10)

where σα with α ∈ {1, 2, 3} are the Pauli matrices (σ± = σ1 ± iσ2 ), a, a∗ denote the annihilation and creation operator, and N = a∗ a is the number operator. The joint domain of all these Hamiltonians is the space D = span{|νi ⊗ |ni | ν ∈ {0, 1} and n ∈ N0 }

(11)

with ν ∈ C2 as canonical basis and |ni ∈ L2 (R) as number basis (Hermite functions). We will assume that the frequencies ωA , ωI and ωC can be controlled independently (or at least two of them) such that we get a control system with control Hamiltonians HJC,j where j ∈ {1, 2, 3} corresponding to the lower half (1 atom) of the energy diagram in Fig. 1 b, where we adopt the widely used convention of forcing the atom (spin) state |↑i to be of ‘higher’ energy than |↓i to compensate for negative Larmor frequencies, see, e.g., the note in [11, p. 144]. The task is to determine the dynamical group G. To this end, we use the strategy described in Subsection 2.5, which follows in this particular case closely the exact solution of the Jaynes-Cummings model [26]. The charge-type operator X1 (determining the block structure) then takes the form X1 = σ3 ⊗ 1 + 1 ⊗ N,

(12)

again with D from (11) as its domain, which in this case turns out to be identical to the space DX1 of finite-particle vectors. The operator X1 is diagonalized by the basis |νi ⊗ |ni. It is convenient to relabel these vectors in order to get |µ, νi = |νi ⊗ |µ − νi ∈ H with µ = n + ν ≥ 0.

(13)

In this basis, we have X1 |µ, νi = µ|µ, νi and the subspaces H(µ) from (4) become H(µ) = span{|µ, 0i, |µ, 1i}

(14)


11

for µ > 0 and H(0) = C|0, 0i for µ = 0. The space DX1 ⊂ H of finite-particle vectors turns out to be identical with the domain D from (11). It is easy to see that the operators HJC,j from Eq. (10) commute with X1 , and therefore we get iHJC,j ∈ u(X1 ). A more detailed analysis, as will be given in Section 4, shows that iHJC,1 and iHJC,2 generate su(X1 ), and therefore we get according to Theorem 2.1: Theorem 3.1. The dynamical group G generated by HJC,j with j ∈ {1, 2} from Eq. (10) coincides with the group SU (X1 ) defined in (6). To get a fully controllable system, apply Theorem 2.4 to see that one has to add a Hamiltonian which breaks the symmetry. A possible candidate is HJC,4 = σ1 ⊗ 1 ∈ B(H).

(15)

If we define the spaces Hα as H− = span{|µ, 0i | µ ∈ N0 }, H0 = {0}, and H+ = span{|µ, 1i | µ ∈ N} the operator HJC,4 becomes complementary to X1 , which can be (µ) (µ) (µ) easily seen since H+ = C|µ, 1i, H− = C|µ, 0i, and H0 = {0}. Hence, according to Thm. 2.3, the control system with Hamiltonians of Eqs. (9,10). H0 = 1, H1 = HJC,1 , H2 = HJC,2 , H3 = HJC,4

(16)

is pure-state controllable+ , and we are recovering a previous result from [39, 35, 34]. However, with our methods we can go beyond this and prove even strong controllability. Thm. 2.4 cannot be applied since dim H(µ) ≤ 2 for all µ, but the analysis of Sect. 5 will lead to an independent argument. Theorem 3.2. The control problem (1) with Hamiltonians Hj and j ∈ {0, . . . , 3} from Eq. (16) is strongly controllable. Hence any unitary U on H can be approximated by varying the control amplitudes u1 = ωA and u2 = ωI in the Hamiltonian HJC of (9) plus flipping ground and excited state of the atom in terms of HJC,4 (with strength u3 )—both in an appropriate timedependent manner. The approximation has to be understood in the strong sense as described in Eq. (2). Finally, note that Theorem 3.2 implies that one can simulate (again in the sense of strong approximations) any unitary V ∈ B(L2 (R)) operating on the cavity mode alone. One only has to find controls uj such that U(0, T ) φ ⊗ ψk ≈ φ ⊗ V ψk for a finite set of states ψk of the cavity (and an arbitrary auxiliary state φ of the atom). 3.2. Many atoms with individual control Next, consider the case of many atoms interacting with the same mode, and under the assumption that each atom (including the coupling with the cavity) can be controlled +

We have omitted the Hamiltonian HJC,3 since it is not needed for the result. However, it can be added as a drift term without changing the result.


12

individually. Such a scenario is relevant for experiments with ion traps, if the number of ions is not too big as have been studied since [40, 41, 42]. The Hilbert space of the system is H = (C2 )⊗M ⊗ L2 (R), (17) where M denotes the number of atoms. We define the basis |bi ⊗ |ni ∈ H where n ∈ N0 , |bi = |b1 i ⊗ · · · ⊗ |bM i, b = (b1 , . . . , bM ) ∈ Z2 × · · · × Z2 = ZM 2 , and the canonical basis 2 |bj i ∈ C with bj ∈ {0, 1}. The control Hamiltonians become HIC,j = σ3,j ⊗ 1 and HIC,M +j = σ+,j ⊗ a + σ−,j ⊗ a∗

(18)

D = span{|bi ⊗ |ni | b ∈ ZM 2 and n ∈ N0 },

(19)

where j ∈ {1, . . . , M} and σα,j = 1⊗(j−1) ⊗ σα ⊗ 1⊗(N −j) . As before, a and a∗ denote annihilation and creation operator. The joint domain of all these operators is

with the basis |bi ⊗ |ni as defined above. As depicted by the red parts in Fig 1, all the HIC,k are invariant under the symmetry defined by the charge operator XM = S3 ⊗ 1 + 1 ⊗ N with S3 =

N X

σ3,j

(20)

j=1

where N = a∗ a denotes again the number operator and D from (19) is the domain of XM . The eigenvalues of XM are µ ∈ N0 and the eigenbasis is given by |µ, bi = |bi ⊗ |µ − |b|i for |b| =

M X j=1

bj ≤ µ.

(21)

In this basis, XM becomes XM |µ, bi = µ|µ, bi and the eigenspaces H(µ) are H(µ) = span{|µ, bi | b ∈ ZM 2 with |b| ≤ µ}. From now on, one may readily proceed as for one atom to arrive at the following analogy to Theorem 3.1: Theorem 3.3. The dynamical group G generated by HIC,k with k ∈ {1, . . . , 2M} from Eq. (18) coincides with group SU (XM ) of unitaries commuting with XM . To get strong controllability, one has to add again one Hamiltonian. As before a σ1 -flip of one atom is sufficient (see the blue parts in Fig 1), and HIC,2M +1 = σ1,1 ⊗ 1.

(22)

is complementary to XM with Hα given by H0 = {0}, H− = span{|µ; 0, b2 , . . . , bM i | µ ∈ N0 , |(b2 , . . . , bM )| ≤ µ}, H+ = span{|µ; 1, b2 , . . . , bM i | µ ∈ N, |(b2 , . . . , bM )| < µ}. Obviously, all the conditions of Thm. 2.4 are satisfied such that one gets Theorem 3.4. The control problem (1) with HIC,k and k ∈ {1, . . . , 2M+1} from (18) and (22) is strongly controllable. As a special case of this theorem, one can approximate any unitary U acting on the atoms alone, i.e. U ∈ U((C2 )⊗M ), by applying Theorem 3.4 to U ⊗ 1. That is, one can simulate U only by operations on one atom and the interactions with the harmonic oscillator. This is used in ion-trap experiments and is known as “phonon bus”.


13

3.3. Many atoms under collective control Now one may modify the setup from the last section by considering again M atoms interacting with one mode, but assuming that one can control the atoms only collectively rather than individually. In other words instead of the Hamiltonians HIC,j and HIC,M +j with j ∈ {1, . . . , M} from Eq. (18) one only has their sums HTC,1 = S3 ⊗ 1 and HTC,2 = S+ ⊗ a + S− ⊗ a∗ ,

where Sα =

PM

j=1 σα,j

(23)

and α ∈ {1, 2, 3, ±}, combinded with the free evolution HTC,3 = 1 ⊗ N

(24)

of the cavity. As before, all operators are defined on the domain D from (19). Note that one readily recovers the original setup from Subsection 3.1 with Pauli operators σα replaced by pseudo-spin operators Sα . The multi-atom analogue of the JaynesCummings Hamiltonian, which can be formed from the HTC,j just defined, is called Tavis-Cummings Hamiltonian [27, 28]. All the Hamiltonians in Eqs. (23) and (24) are invariant under the U(1)-action generated by XM of Eq. (20). However, this is not the only symmetry, since all these HTC,j are also invariant under the permutation of the atoms. Therefore, one may no longer exhaust the group SU (XM ) as in Theorem 3.3 (since the following operators cannot be reached: those commuting only with XM but not also with permutations of the atoms). A minimal modification is to restrict the states of the atoms to spaces on which permutation-invariant unitaries operate transitively∗ . The most natural choice 2 ⊗M is the symmetric tensor product (C2 )⊗M , i.e. the Bose subspace of (C2 )⊗M . sym ⊂ (C ) ⊗ν The preferred basis of (C2 )⊗M ⊗ |0i⊗(M −ν) with ν ∈ {0, . . . , M} sym is |νi = SymM |1i and the projection SymM from (C2 )⊗M onto the symmetric subspace (C2 )⊗M sym . In other words |νi is the unique, pure, permutation-invariant state with ν atoms in the excited state |1i and M−ν ones in the ground state |0i. Therefore, (C2 )⊗M sym can be identified M +1 with the Hilbert space C of a (pseudo-)spin-M/2 system. Its basis |νi, with ν ∈ {0, . . . , M} becomes the canonical basis. Combining this with L2 (R) for the cavity one gets Hsym = CM +1 ⊗ L2 (R) as the new Hilbert space of the system. All the operators defined above (HTC,1 , HTC,2 , HTC,3 and XM ) can be restricted to Hsym (and in slight abuse of notation we will re-use the symbols after restriction) and their domain becomes Dsym = span{|νi ⊗ |ni | ν ∈ {0, . . . , M} and n ∈ N0 },

(25)

which is just the projection of D from (19), i.e. Dsym = SymM D. The eigenbasis of XM now takes the form |µ, νi = |νi ⊗ |µ − νi where µ ∈ N0 and ν < dµ = min(µ, M+1). For the XM -eigenspaces, we get again XM |µ, νi = µ|µ, νi and (µ) Hsym = span{|µ, νi | ν ∈ {0, . . . , dµ }}.

(26)

∗ An alternative strategy would be to treat permutation symmetry in the same way as U(1)-symmetry. However, already the restriction to permutation-invariant states will turn out to be difficult enough.


14

Now one can proceed as the in the previous cases: The operators HTC,1 , HTC,2 , HTC,3 are (as operators on Hsym ) invariant under the action generated by XM and therefore elements of u(XM ). However, one still cannot exhaust all of U(XM ) (or SU (XM )). One only gets: Theorem 3.5. The dynamical group G generated by the operators HTC,1 , HTC,2 , HTC,3 from Eqs. (23) and (24) is a strongly closed subgroup of U(XM ). For each unitary V ∈ U(XM ) and each µ ∈ N0 we can find an element U ∈ G such that Uψ (µ) = V ψ (µ) (µ) holds for all ψ (µ) ∈ Hsym . In other words: As long as the charge µ is fixed, one can still approximate any V ∈ U(XM ), but if one considers superpositions of different charges this is no longer the case, i.e. there are ψ ∈ DXM and V ∈ U(XM ) such Uψ 6= V ψ for all U ∈ G. We have checked the latter explicitly with the computer algebra system Magma [43] for the case M = 2. To circumvent this problem, one has to add control Hamiltonians. Unfortunately, it seems that one has to add quite a lot. The best result we have got so far is to replace the operators from Eqs. (23) and (24) by HCC,k = |kihk| − |k−1ihk−1| ⊗ 1 with k ∈ {1, . . . , M}, HCC,M +1 = HTC,2 = S+ ⊗ a + S− ⊗ a∗ and HCC,M +2 = |0ih1| + |1ih0| ⊗ 1. (27) The operators HCC,k with k ∈ {1, . . . , M +1} commute with XM and generate (as we will see in Sect. 4.4) the Lie algebra su(XM ). In addition we have HCC,M +2 which is complementary to XM with Hilbert spaces H+ = span{|µ; 0i | µ ∈ N0 }, H− = span{|µ; 1i | µ ∈ N}, and H0 = span{|µ, νi, | µ ∈ N, µ > 2, ν ∈ {3, . . . , min(M, µ)}}. Note that we get an example for Def. 2.2 with a non-trivial H0 . Now one can apply Thms. 2.1 and 2.4 to get the analogues of Theorems 3.1 and 3.2: Theorem 3.6. The dynamical group G generated by HCC,k with k ∈ {1, . . . , M+1} from Eq. (27) coincides with the group SU (XM ) of unitaries commuting with XM .

Theorem 3.7. The control problem (1) with H0 = 1 and HCC,k for k ∈ {1, . . . , M+2} from (27) is strongly controllable. To be able to control all diagonal traceless operators HCC,k , with k ∈ {1, . . . , M} is a very strong assumption. Unfortunately, a detailed analysis including computer algebra indicates that we cannot recover Theorem 3.7 with fewer resources. 4. A Lie algebra of block-diagonal operators The purpose of this section is to re-discuss abelian symmetries and to provide technical details (in particular proofs) we omitted in Sections 2 and 3. To this end, re-use the notations already introduced in Section 2.5. In particular, the abelian symmetry induces a block-diagonal decomposition which, in infinite dimensions, allows for defining a block-diagonal Lie algebra and its exponential map onto a block-diagonal Lie group; see Propositions 4.1 and 4.2. We identify the set of all block-diagonal unitaries reachable by


15

block-diagonal time evolutions in Proposition 4.4 as the strong closure of exponentials of block-diagonal Lie algebra elements. A central result is Corollary 4.6, in which the question of controllability for the block-diagonal system of infinite dimensions is reduced to analyzing controllability for all finite-dimensional blocks. Using finite-dimensional commutator calculations one can now establish controllability on the infinite-dimensional but block-diagonal space for each of the three control systems analyzed. 4.1. Commuting operators The first step is a closer look at the Lie algebra u(X) and the corresponding group U(X) introduced in Theorem 2.1 (which we will prove in this context). To this end, let us start with a unitary U commuting with the representatives π(z), i.e. [π(z), U] = 0 for all P (µ) (µ) z ∈ U(1). This is equivalent to Uψ = ∞ ψ for all ψ ∈ H with ψ (µ) := X (µ) ψ ∈ µ=0 U H(µ) given a sequence of unitaries U (µ) on the µ-eigenspaces H(µ) of X. Similarly one can consider a selfadjoint H with domain D(H) commuting with X. By definition♯ this means the spectral projections of H commute with the X (µ) , which is equivalent to DX ⊂ D(H), HDX ⊂ DX and Hψ =

∞ X µ=0

H (µ) ψ (µ) for ψ ∈ DX

(28)

with a sequence of selfadjoint operators H (µ) on the eigenspaces H(µ) and the ψ (µ) as defined above. The H(µ) are finite-dimensional, and therefore the H (µ) are bounded. Hence the unboundedness of H is inherited only from the unboundedness of the sequence of norms kH (µ) k. So it is easy to see that all elements of DX are analytic for H and therefore DX becomes a domain of essential selfadjointness for H (i.e. H is uniquely determined by its restriction to DX as a consequence of Nelson’s analytic vector theorem [44, Thm. X.39]). Accordingly, we will denote (in slight abuse of notation) the selfadjoint operator H and its restriction to DX by the same symbol. This proves very handy when introducing, on the set u(X) of anti-selfadjoint operators commuting with X, the structure of a Lie algebra by (λQ1 + Q2 )ψ = λQ1 ψ + Q2 ψ, [Q1 , Q2 ]ψ = Q1 Q2 ψ − Q2 Q1 ψ for Q1 , Q2 ∈ u(X), λ ∈ R, and ψ ∈ DX . The linear combination λQ1 + Q2 and the commutator [Q1 , Q2 ] are defined only on the joint domain DX but since they are essentially selfadjoint on it, their selfadjoint extensions exist and are uniquely determined. This proves the first statement of Thm. 2.1, which we restate as follows: Proposition 4.1. A selfadjoint operator H commuting with X admits DX as an invariant domain of essential selfadjointness. The space u(X) = iH | H = H ∗ commuting with X (29) P (µ) (µ) = iH | Hψ = µ H ψ , ψ ∈ DX , H (µ) = (H (µ) )∗ ∈ B(H(µ) ) (30) becomes a Lie algebra with the commutator as its Lie bracket. ♯ Note that the identity [X, Y ] ψ = 0 for all ψ on a common dense domain is–in contrast to popular belief–not a proper definition for two commuting selfadjoint operators; cf. the discussion in [36, VIII.6]. Fortunately, such pathological cases do not occur in our set-up.


16

Since all iH ∈ u(X) are anti-selfadjoint, they admit a well-defined exponential map exp(iH). Boundedness of the H (µ) together with Eq. (28) allows to express exp(iH) very explicitly. More precisely one has exp(iH) ψ =

∞ X µ=−∞

exp(iH (µ) ) ψ (µ) where ψ (µ) = X (µ) ψ ∈ H(µ)

(31)

P (µ) n ) /(n!). This shows that exp : u(X) → U(X) is welland exp(iH (µ) ) = ∞ n=0 (iH defined and onto as stated in Thm. 2.1, which we are now ready to prove: Proposition 4.2. The exponential map on u(X) is well-defined and given in terms of Equation (31). It maps u(X) onto the strongly closed subgroup U(X) = U ∈ U(H) | [U, π(z)] = 0 for all z ∈ U(1) P = U | Uψ = µ U (µ) ψ (µ) , ψ ∈ H, U (µ) ∈ U(H(µ) ) of U(H).

(32) (33)

Proof. The only statement not yet proven is the closedness of U(X). To this end, we have to show that for any net (Uλ )λ∈I strongly converging to a bounded operator U we have U ∈ U(X). As Uλ ∈ U(X) we have [π(z), Uλ ] = 0 for all λ. Due to strong continuity of the map A 7→ [π(z), A] and the convergence of the Uλ to U it follows P that [π(z), U] = 0. Hence U decomposes into a strongly converging series U = µ U (µ) (µ)

with U (µ) ∈ B(H(µ) ), and for each fixed µ we get limλ Uλ = U (µ) . Since H(µ) is finite(µ) dimensional, the nets (Uλ )λ∈I converge in norm and therefore U (µ) ∈ U(H(µ) ) which implies U ∈ U(X).

Note that we actually proved more than what we stated. A strongly convergent sequence (or net) of elements of U(X) cannot converge to an isometry which is not unitary as well. Hence U(X) is strongly closed as a subset of B(H)—and not only as a subset of U(H) as generally is the case (cf. corresponding remarks in Sect. 2.4). The remaining statements in this subsection are devoted to the dynamical group G generated by a family of selfadjoint operators H1 , . . . , Hd . Recall that we have introduced it as the smallest strongly closed subgroup of U(H) containing all unitaries of the form exp(itHk ). If the Hk are commuting with X, i.e. iHk ∈ u(X), then the group G is a subgroup of U(X), and the simple structure of the latter makes explicit calculations at least feasible. In the following, we show how U(X) is related to the Lie algebra l generated by the iHk . To this end, we need some additional notations. For each K ∈ N, U ∈ U(X), and iH ∈ u(X), let us consider U

[K]

=

K X µ=0

U

(µ)

, H

[K]

=

K X µ=0

H

(µ)

, H

[K]

=

K M µ=0

H(µ) .

(34)

The operators U [K] and H [K] act on the finite-dimensional Hilbert space H[K] . Therefore all operator topologies coincide and we can apply the well-known finite-dimensional [K] theory. The dynamical group G [K] (generated by Hk with k ∈ {1, . . . , d}) becomes a


17

closed subgroup of the unitary group U(H[K] ), which is a Lie group. Hence G [K] is a Lie [K] group, too, and its Lie algebra l[K] is generated by iHk with k ∈ {1, . . . , d}. Now, the crucial point is that one can approximate the infinite-dimensional objects G and l by the finite-dimensional G [K] and l[K]. To see this, the first step is the following lemma. Lemma 4.3. Consider the Lie algebras l ⊂ u(X) and l[K] ⊂ B(H[K] ) (with K ∈ N) [K] [K] ˜ ∈ l[K] can generated by iH1 , . . . , iHd and iH1 , . . . , iHd , respectively. Each element Q ˜ = Q[K] for some element Q ∈ l. be written as Q P [K] [K] ˜ ∈ l[K] , it is equal to a linear combination Proof. Since Q ℓ cℓ Cℓ (iHj1 , . . . , iHjn ) of [K] [K] [K] [K] repeated commutators Cℓ (iHj1 , . . . , iHjn ) containing the elements {iHj1 , . . . , iHjn } with jk ∈ {1, . . . , d}. However, l is generated by iH1 , . . . , iHk and it contains the same [K] commutators Cℓ (iHj1 , . . . , iHjn ) yet with Hj replaced by Hj . Hence one can form a ˜ as stated. linear combination Q such that Q[K] = Q Moreover, we now have the tools to prove the relation between the Lie algebra l and the dynamical group G already stated in Thm. 2.1. Proposition 4.4. Consider again iH1 , . . . , iHd ∈ u(X) and the Lie algebra l generated by them. Then the corresponding dynamical group G coincides with the strong closure of exp(l) ⊂ U(X). Proof. Each U ∈ G can be written as the limit of a net (Uλ )λ∈I of operators Uλ , which are monomials in exp(itk Hk ) with k ∈ {1, . . . , d} with appropriate times tk . This implies in particular that the Uλ commute with π(z) for all z, and, by continuity, the same is true for U. Hence U ∈ U(X), and for each K ∈ N we can define U [K] which is the limit [K] of the net (Uλ )λ∈I . The latter converges in norm (since H[K] is finite-dimensional), and therefore U [K] ∈ G [K] . This implies U [K] = exp(QK ) with QK ∈ l[K] as G [K] is a Lie group and l[K] its Lie algebra. For U to be in the strong closure of exp(l), each strong ǫ-neighborhood of U, i.e. the sets N (U; ψ1 , . . . , ψf ; ǫ) introduced in Eq. (3), should contain an element of exp(l) for all ψ1 , . . . , ψf and all ǫ > 0. However, the unitary group is contained in the unit ball of B(H), and thus it is sufficient to consider only those N (U; ψ1 , . . . ; ψf , ǫ) with vectors ψ1 , . . . , ψf from a dense subspace of H; cf. [45, I.3.1.2]. Hence, in turn, it is sufficient to consider only neighborhoods with ψj ∈ DX . But then there is a K ∈ N such that ψj ∈ H[K] for all j ∈ {1, . . . , f }. Now take the operator QK from the last ˜ K ∈ l with Q ˜ [K] = QK , which exists due to Lemma 4.3. By construction paragraph and Q K ˜ K )]ψj k = k[U [K] − exp(Q ˜ K )[K]]ψj k = k[(U [K] − exp(Q ˜ [K] )]ψj k = we have k[U − exp(Q K k[(U [K] − exp(QK )]ψj k = 0 since U [K] = exp(QK ), as was also seen in the previous ˜ K ) ∈ N (U; ψ1 , . . . , ψf ; ǫ) which shows that U is in the strong paragraph. Hence exp(Q closure of exp(l). This shows that the dynamical group G is contained in the strong closure of exp(l). Conversely, consider exp(Q) for Q ∈ l. We have to show that exp(Q) is in the dynamical group G. To this end we observe, for each K ∈ N, that exp(Q[K] ) = exp(Q)[K] ,


18 [K]

[K]

which is obviously in G [K] . Hence there is a UK = exp(iHj1 ) · · · exp(iHjn ) with jk ∈ {1, . . . , d} which is ǫ-close (in norm) to exp(Q[K] ). As in the last paragraph, this implies ˜ = exp(iHj1 ) · · · exp(iHjn ) is in N (exp(Q); ψ1 , . . . , ψf ; ǫ) provided ψj ∈ H[K] for that U all j ∈ {1, . . . , f }. Hence exp(Q) is in the strong closure of the group of monomials in the exp(iHj ), but this is just the dynamical group G. Since G is strongly closed, the strong closure of exp(l) is contained in G, too. Since we have shown the other inclusion before, the entire proposition is proven. Moreover, with this proposition the proof of Thm. 2.1 is complete. – The rest of this subsection is devoted to analyzing a related question: If, in finite dimension, two Lie algebras l1 , l2 generate the same group, then they are actually identical. However, in infinite dimensions this no longer true. Therefore, the next proposition is meant to decide if dynamical groups generated by two different sets of Hamiltonians do in fact coincide. Proposition 4.5. Consider two Lie algebras l1 , l2 ⊂ u(X). Assume that for each Q ∈ l1 ˜ ∈ l2 such that Q[K] = Q ˜ [K] holds (note that we can have and each K ∈ N, there is a Q ˜ for the same Q but different K). Then exp(l1 ) is contained in the strong different Q closure of exp(l2 ). Proof. One may readily use the same strategy as in the proof of Prop. 4.4: If the given condition holds, one can find in each neighborhood N (exp(Q); ψ1 , . . . , ψf ; ǫ) of exp(Q) ˜ with Q ˜ ∈ l2 . Hence exp(Q) is in the strong closure of with ψ1 , . . . , ψf ∈ DX an exp(Q) exp(l2 ). Inserting su(X) for l2 provides a useful criterion to check whether the dynamical group G generated by H1 , . . . , Hd ∈ su(X) is as large as possible in the sense that G = SU (X). To this end, let us introduce the truncated versions (µ) su[K](X) = {Q[K] | Q ∈ su(X)} = ⊕K ), µ=0 su(H

(µ) SU [K](X) = {U [K] | U ∈ SU (X)} = ⊕K ), µ=0 SU (H

(35)

where we have used for any finite-dimensional subspace K of H the notations su(K) for the Lie algebra of traceless operators on K and similarly SU (K) for the Lie group of unitaries on K with determinant 1. Note that elements of su(K) and SU (K) have—as operators on H—a finite rank and their support and range are both contained in K. Corollary 4.6. Consider Hamiltonians iH1 , . . . , iHd ∈ su(X), the corresponding dynamical group G and the generated Lie algebra l. If su[K](X) = l[K] holds for all K ∈ N, then one finds G = SU (X). Proof. Simple application of Props. 4.4 and 4.5.


19

4.2. One atom The material just introduced readily applies to the systems studied in Sect. 3. This includes in particular the proofs of Thms. 3.1, 3.3, 3.5 and 3.6. The first step is again one atom interacting with a cavity (Sect. 3.1). Hence the Hilbert space is H = C2 ⊗L2 (R) and the U(1)-symmetry under consideration is generated by the operator X1 = σ3 ⊗ 1 + 1 ⊗N already defined in (12). The domain of X1 is D from Eq. (11), which is identical to DX1 introduced in (5). The next step is to characterize the Lie algebra l generated by the control Hamiltonians HJC,1 and HJC,2 as defined in (10). They admit D = DX1 as a joint common domain, and it is easy to see that they commute with X1 (in the sense introduced in the previous subsection). Hence l ⊂ u(X1 ), and all the machinery from Subsection 4.1 applies. This includes in particular the block-diagonal decomposition of operators A ∈ u(X1 ) given in Eq. (28). In our case the subspaces H(µ) with µ ∈ N are given by (cf. Eq. (14)) H(µ) = span{|µ, 0i, |µ, 1i} using the basis |µ, νi ∈ H introduced in (13). For µ = 0, we (µ) get the one-dimensional space H(0) = C|0, 0i. The restrictions HJC,j of the operators HJC,j to the subspaces H(µ) are given by (for µ ≥ 1): √ (µ) (µ) (µ) (µ) (µ) (µ) HJC,1 = −ς3 /2, HJC,2 = µς1 , HJC,3 = (µ + 1/2)ς0 − ς3 /2, (36) P (µ) where we have introduced the operators ςα = with α ∈ {0, . . . , 3} via their µ ςα

projections ς0 = 1(µ) = X (µ) = |µ, 0ihµ, 0| + |µ, 1ihµ, 1|, ς1 = |µ, 0ihµ, 1| + |µ, 1ihµ, 0|, (µ) (µ) ς2 = i |µ, 1ihµ, 0| − |µ, 0ihµ, 1| , and ς3 = |µ, 0ihµ, 0| − |µ, 1ihµ, 1|. Hence, for each (µ) fixed µ, the operator ςα is just the corresponding Pauli operator on H(µ) given in the basis |µ, 0i, |µ, 1i. We have used the core symbol ς rather than σ in order to avoid confusion with the operators σα ⊗ 1 acting only on the atom. In addition we introduce the operators Aα,k ∈ u(X1 ) with α ∈ {0, . . . , 3} and k ∈ N0 by p Aα,k = X1 X1k ςα for α ∈ {1, 2}, A3,k = X1k ς3 , A0,k = X1k . (37) (µ)

(µ)

In terms of the Aα,k , now the HJC,j can readily be re-expressed as HJC,1 = −A3,0 /2, HJC,3 = A1,0 /2, HJC,2 = A0,1 + (A0,0 − A3,0 )/2.

(38)

The next lemma shows that the Lie algebra l generated by the HJC,j is spanned as a vector space by a subset of the Aα,k . Lemma 4.7. The Lie algebra l generated by iHJC,j with j ∈ {1, 2} is spanned as a vector space by the operators iAα,k with α ∈ {1, 2, 3} and k ∈ N0 . Proof. Obviously the operators iAα,k are in su(X1 ). Hence, they span a subspace ˜l ⊂ su(X1 ). To prove that ˜l is a Lie subalgebra of su(X1 ) one only has to check that [Aα,k , Aβ,j ] ∈ ˜l for all α, β ∈ {1, 2, 3} and j, k ∈ N0 . This follows easily, because the Aα,k are just products of powers of X1 and the ςα . But the latter are representatives of the Pauli operators. Hence [A1,k , A2,ℓ ] = 2iA3,k+ℓ+1 , [A3,k , A1,ℓ ] = 2iA2,k+ℓ , [A2,k , A3,ℓ ] = 2iA1,k+ℓ ,

(39)


20

All operators vanish in the case of µ = 0. Hence ˜l is a Lie algebra and Eq. (38) proves that l ⊂ ˜l. For proving ˜l = l, one has to express the Aα,k for α ∈ {1, 2, 3} and k ∈ N0 in terms of repeated commutators of the HJC,2 and HJC,3 . By the commutation relations in Equation (39) it is obvious that ˜l is generated (as a Lie algebra) by Aα,0 with α ∈ {1, 2, 3}. Therefore, the statement follows from Eq. (38), which in turn shows that A1,0 and A3,0 are just HJC,3 and HJC,1 , while A2,0 can be derived from the commutator [HJC,1 , HJC,3 ]. With this Lemma and the material developed in the last subsection, one can proceed to determine the structure of the dynamical group generated by HJC,1 and HJC,2 . This is the content of Thm. 3.1, which is restated (and proven) here as a proposition. Proposition 4.8. The dynamical group generated by HJC,1 and HJC,2 is equal to SU (X). Proof. According to Prop. 4.4 the dynamical group G is the strong closure of exp(H) with H ∈ l, i.e. the Lie algebra generated by HJC,1 and HJC,2 , while SU (X) is the strong closure of exp(su(X)). Hence, by Cor. 4.6 we have to show that the truncated algebras l[K] and su[K](X) are identical. The inclusion l[K] ⊂ su[K](X) is trivial, since all the (µ) blocks HJC,j with j ∈ {1, 2} are traceless. To show the other inclusion, first note that l[0] = su(X)[0] = {0}. Hence it is sufficient to check that for each fixed 0 < µ0 ≤ K and each iH ∈ su[K] (X) with H (µ) = 0 for µ 6= µ0 there is an iA ∈ l such that iA(µ0 ) = iH (µ0 ) and A(µ) = 0 for all 0 < µ ≤ K with µ 6= µ0 . The rest follows by linearity. For constructing such an A, recall from Lemma 4.7 that l is spanned (as a vector space) by the Aα,k with α ∈ {1, 2, 3} and k ∈ N0 . Now consider a polynomial f in one real variable satisfying√f (µ) = 0 for all 0 < µ ≤ K with µ 6= µ0 and f (µ0 ) = 1. The operators Bα,f = f (X) Xςα with α ∈ {1, 2} and B3,f = f (X)ς3 are linear combinations (µ) of the Aα,k , and they satisfy the condition Bα,f = 0 for all 0 < µ ≤ K such that µ 6= µ0 √ (µ ) (µ ) and Bα,f0 = cα ςα 0 for a constant cα given by c1 = c2 = µ0 and c3 = 1. But all (µ )

traceless operators H (µ0 ) ∈ B(H(µ0 ) ) can be written as a linear combinations of the ςα 0 , which concludes the proof. Before proceeding to the next subsection, consider the free Hamiltonian of the cavity HJC,3 . We have omitted it from the discussion of the dynamical group, and the reason can be seen easily from (39): HJC,2 differs from HJC,1 only by X1 + 1/2 which commutes with all elements of su(X). Hence adding HJC,3 as a control Hamiltonian would just add a one-dimensional center to the dynamical group G = SU (X). For the same reason, HJC,3 could be easily added as a drift term. Any effect it may have can be undone by evolving the system with HJC,1 , and the remaining relative phase between sectors of different charge µ does not affect the discussion of strong controllability in Sect. 5. Finally, let us remark that—due to the same reasons just discussed—we could exchange HJC,1 and HJC,3 almost without changes to the results of this subsection.


21

4.3. Many atoms with individual control First, recall some notations from Sect. 3.2. The Hilbert space is HM = (C2 )⊗M ⊗ L2 (R) using the distinguished basis |µ; ~bi with ~b ∈ ZM 2 from Eq. (21). The charge operator is XM = S3 ⊗ 1 + 1 ⊗ N, cf. Eq. (20), with domain DM from Eq. (19). In addition, let us introduce the re-ordered tensor product (where |µ, b1 , . . . , bM i ∈ HM and b ∈ Z2 ) ˆ k |bi = |µ + b; b1 , . . . , bk−1 , b, bk , . . . , bM i ∈ HM +1 . |µ, ~bi⊗

(40)

The key result of this section is split into the following three lemmas, which eventually will lead to a proof of Thm. 3.3. (µ)

(µ)

Lemma 4.9. The complexification suC (HM ) of the real Lie algebra su(HM ) is generated ~ c. by elements |µ; ~bihµ; ~c| with ~b, ~c ∈ ZM 2 satisfying b 6= ~ (µ)

(µ)

Proof. suC (HM ) is isomorphic to the Lie algebra sl(HM ) of traceless operators on (µ) (µ) HM . The |µ; ~bihµ; ~c| with ~b 6= ~c span the vector space of all A ∈ B(HM ) satisfying hµ; ~b | A | µ; ~bi = 0 for all ~b ∈ ZM 2 i.e. all operators which are off-diagonal in the basis (µ) ~ |µ; bi. The smallest Lie algebra containing this space is sl(HM ). (µ)

Lemma 4.10. The Lie algebra suC (HM +1 ) is generated by the union of the subalgebras (µ−b) ˆ k |bi) with b ∈ Z2 and k ∈ {1, . . . , M}. suC (HM ⊗ (µ−b) ˆ k |bi ∈ Proof. First of all, note that (by definition) |µ−b; ~bi ∈ HM . Hence |µ−b; ~bi⊗ (µ) (µ) ˆ k |bi are subspaces of HM HM +1 which shows that all the Hilbert spaces H(µ−b) ⊗ . ~ According to the previous lemma, we have to show that operators A = |µ; bihµ; ~c| +1 with ~b, ~c ∈ ZM and ~b 6= ~c can be written as commutators from operators in 2 (µ+b) ˆ k |bi). We have to distinguish two cases: In the first case, there the suC (HM ⊗ is at least one k ∈ {1, . . . , M} with bk = ck = b. If this holds, A can be written as |µ−b; b1 , . . . , bk−1 , bk+1 , . . . , bM +1 ihµ−b; c1 , . . . , ck−1 , ck+1, . . . , cM +1 | ⊗ |bihb| ∈ (µ−b) ˆ k |bi). The second case arises if bk 6= ck for all k. Now consider the commusuC (HM ⊗ tator of the operators B = |µ; ~bihµ; b1 , c2 , . . . , cM +1 | and C = |µ; b1 , c2 , . . . , cM +1 ihµ; ~c| (µ−b ) (µ−ck ) ˆ 1 |b1 i), and C ∈ suC (HM ˆ k |ck i) for k > 1. obviously A = [B, C], B ∈ suC (HM 1 ⊗ ⊗ This concludes the proof. (µ)

Lemma 4.11. The Lie algebra su(HM +1 ) is contained in the Lie algebra g generated by (µ−1) (µ) ˆ k 1. ˆ k 1 and su(HM )⊗ su(HM )⊗ Proof. First of all note that it is sufficient to prove the statement for the corresponding (µ) (µ) (µ) complexified Lie algebras suC (HM +1 ) = su(HM +1 ) ⊕ i(HM +1 ) and gC = g ⊕ ig, since (µ) we get the original statement back by restricting the inclusion suC (HM +1 ) ⊂ gC to anti-selfadjoint elements on both sides. ˆ k |0ih0| + a⊗ ˆ k |1ih1| with ˆ k 1 are of the form A = a⊗ The elements of suC (H(µ) )⊗ (µ) ˆ k |bihb| ∈ gC a ∈ suC (H ). We will show that both summands are elements of gC , i.e. a⊗ for b ∈ {0, 1}. The same holds for µ−1. The statement then follows from Lemma 4.10.


22

~ Use again Lemma 4.9 and choose a = |µ; ~bihµ; ~c| with ~b, ~c ∈ ZM c. We 2 and b 6= ~ ˆ k |0ih0| + a⊗ ˆ k |1ih1| as rewrite A = a⊗ |µ; (b1 , . . . , bk , 0, bk+1 , . . . , bM )ihµ; (c1, . . . , ck , 0, ck+1, . . . , cM )|

+|µ + 1; (b1 , . . . , bk , 1, bk+1 , . . . , bM )ihµ + 1; (c1 , . . . , ck , 1, ck+1, . . . , cM )|.

(41)

Moreover, ~b0 := (b2 , . . . , bk , 0, bk+1, . . . , bM ), ~b1 := (b2 , . . . , bk , 1, bk+1 , . . . , bM ), ~c0 := (c2 , . . . , ck , 0, ck+1, . . . , cM ), and ~c1 := (c2 , . . . , ck , 1, ck+1, . . . , cM ) allows us to simplify ˆ 1 |b1 ihc1 |. A = (|µ−b1 ; ~b0 ihµ−c1 ; ~c0 | + |µ−b1 +1; ~b1 ihµ−c1 +1; ~c1 |)⊗

(42)

ˆ 11 Next, consider a second operator B = (|µ−c1 ; ~c0 ihµ−c1 ; ~c0 | − |µ−c1 ; ~c1 ihµ−c1 ; ~c1 |)⊗ and assume that M > 1 holds. Then there is a ℓ ∈ {1, . . . , M} with bℓ 6= cℓ . Without loss ˆ j |bj ihcj | with of generality one can assume that ℓ 6= 1 (otherwise rewrite A in (42) as A˜⊗ ~ ˆ 1 |b1 ihc1 | = another index j). The commutator now equals [A, B] = |µ−b1 ; b0 ihµ−c1 ; ~c0 |⊗ ˆ k |0ih0|. If M = 1 one has two possible cases: either b = 0 and c = 1 or b = 1 and a⊗ c = 1. In the first case choose B = (|µ−c; 0ihµ−c; 0| − |µ−c; 1ihµ−c; 1|) ⊗ 1, and in the second case pick B = (|µ−b; 0ihµ−b; 0| − |µ−b; 1ihµ−b; 1|) ⊗ 1. Then the commutator [A, B] leads again to ±|µ−b; 0ihµ−c; 0| ⊗ |bihc|. (µ) ˆ k |0i) ⊂ gC for all k. The same reasoning Therefore, one can conclude that suC (HM ⊗ (µ−1) ˆ k |1i). Hence the statement follows from the previous lemma. holds for suC (HM ⊗ Now let us consider the control Hamiltonians HIC,j , HIC,M +j from Equation (18). We will use Lemma 4.11 and an induction in M to prove Thm. 3.3, which we restate here as a proposition. Proposition 4.12. The dynamical group generated by the control Hamiltonians HIC,j with j ∈ {1, . . . , 2M} is identical to SU (XM ). Proof. According to Corollary 4.6 we have to show that for each K, we find that [K] lM = su[K] (XM ), where lM denotes the Lie algebra generated by the HIC,j with j ∈ {1, . . . , 2M}. Since lM ⊂ su(XM ) is trivial, only the other inclusion has to be shown. This will be done by induction. By Prop. 4.8 the statement is true for M = 1. Now we assume it is true for M to show that it is true for M+1, too. To this end, consider for each k ∈ {1, . . . , M+1} the Hamiltonians HIC,j , HIC,M +1+j with j ∈ {1, . . . , M+1} and j 6= k. They can be regarded as operators on the Hilbert space HM and they generate a Lie algebra lM which satisfies by assumption [K] lM

[K]

= su

(XM ) =

K M

(µ)

su(HM )

(43)

µ=1

ˆ k 1 ⊂ lM +1 and for all K. As operators on HM +1 , they generate the Lie algebra lM ⊗ [K+1] (µ) ˆ k 1 ⊂ lM +1 holds for all µ ≤ K and k ∈ according to (43) one finds that su(HM )⊗ (µ) {1, . . . , M+1}. Thus, we can apply Lemma 4.11 and su(HM +1 ) is contained in the Lie [K+1] (K) (K) algebra lM +1 for all µ ≤ K. But since lM +1 ⊂ su[K] (XM +1 ) = su(HM +1 ), one even gets [K] su[K](XM +1 ) ⊂ lM +1 , just as was to be shown.


23

4.4. Many atoms under collective control As a last topic in this section, we provide proofs for Thms. 3.5 and 3.6. To this end, recall the notation from Sect. 3.3. The Hilbert space is Hsym = CM +1 ⊗ L2 (R) with basis |µ; νi = |νi ⊗ |µ−νi where ν ∈ {0, . . . , dµ } and dµ = min(µ, M).

(44)

The charge operator is again XM = S3 ⊗ 1 + 1 ⊗ N from Eq. (20) but now as an (µ) operator on Hsym with domain Dsym defined in (25) and the µ-eigenspaces Hsym become (µ) Hsym = span{|µ; νi | ν ∈ {0, . . . , dµ }}; cf. Eq. (26). The control Hamiltonians are HTC,j with j ∈ {1, . . . , 3} defined in (23) and (24). In addition let us introduce the operators Y3 , Y± ∈ suC (XM ) (which denotes again the complexification of su(XM )) given by dµ −1

Y3 |µ; νi = ν|µ; νi,

(µ) Y+

=

X ν=0

|µ; ν+1ihµ; ν|,

(µ) Y−

=

dµ X ν=1

|µ; ν−1ihµ; ν|.

(45)

They are related to the HTC,j by HTC,1 = Y3 − (M/2) 1, HTC,3 = XM − Y3 ,

HTC,+ = S+ ⊗ a = f (XM , Y3 )Y+ , HTC,− = S− ⊗ a∗ = Y− f (XM , Y3 ) where f is a function in two variables x, y given by p p √ f (x, y) = h1 (x, y)h2 (y) y, h1 (x, y) = x + 1 − y, h2 (y) = M + 1 − y,

(46)

(47)

and f (XM , Y3) has to be understood in the sense of functional caculus (both operators (µ) commute). As operators on Hsym for fixed µ, the Y± satisfy Y+ Y− = 1 − |µ, 0ihµ, 0|, Y− Y+ = 1 − |µ, dµ ihµ, dµ|

(48)

and for any function g(y) which is continuous on the spectrum of Y3 , one finds Y+ g(Y3) = g(Y3 − 1)Y+ , Y− g(Y3) = g(Y3 + 1)Y− .

(49)

We are now prepared for the first lemma. Lemma 4.13. The operators HTC,1 , HTC,+ = S+ ⊗ a, and HTC,− = S− ⊗ a∗ satisfy the following commutation relations (as operators on H(µ) ) (i) [Y3n−1 HTC,+ , HTC,− ] = Pn−1 n k Y3 and (ii) [Y3n+1 , HTC,+ ] = (XM −Y3 )Y3n +(N 1 −Y3 )Y3n −(XM −Y3 )(N 1 −Y3 ) k=0 k Pn n n−k k Y3 HTC,+ . k=0 k (−1) Proof. Using Eq. (46) to re-express HTC,± in terms of Y± , Y3 and XN , we get for the first commutator [Y3n−1 HTC,+ , HTC,− ] = Y3n−1 f (XM , Y3 )Y+ Yf (XM , Y3 ) − Y− f 2 (XM , Y3 )Y3n−1 Y+ .

(50)

It is easy to check that f (XM , Y3 )|µ; 0i = 0 holds. Together with (48) this leads to Y3n−1 f (XM , Y3)Y+ Y− = Y3n−1 f (XM , Y3 ).

(51)


24

With (49) we get on the other hand Y− f 2 (XM , Y3 )Y+ = f 2 (XM , Y3 + 1)(Y3 + 1)n−1 Y− Y+ . Now observe that h21 (XM , Y3 + 1)h22 (Y3 + 1)|µ; dµ i = 0 and use again (48) to get Y− f 2 (XM , Y3 )Y+ = f 2 (XM , Y3 + 1)(Y3 + 1)(n−1) .

(52)

Inserting (51) and (52) into (50) leads to [Y3n−1 HTC,+ , HTC,− ] = Y3n−1 f 2 (XM , Y3 ) − f 2 (XM , Y3 + 1)(Y3 + 1)n−1, where we have used the fact that f (XM , Y3) and Y3 commute. Inserting the definition of f in (47) and expanding (Y3 + 1)n−1 leads to the first commutator. The second commutator follows similarly from [Y3n+1 , HTC,+ ] = Y3n+1 f (XM , Y3 )Y+ − f (XM , Y3)Y+ Y3n+1 and applying (49) to commute Y+ to the right. We are now ready to prove Thm. 3.5. The statement about the dynamical group G as a subgroup of U(XM ) is an easy consequence of the discussion in Sect. 4.1. The second statement in Thm. 3.5 is rephrased in the following Proposition. Proposition 4.14. Consider the Lie algebra lTC ⊂ u(XM ) generated by the HTC,j with (µ) (µ) j ∈ {1, . . . , 3} and µ ∈ N. The restriction lTC of lTC to Hsym coincides with the Lie (µ) (µ) algebra u(Hsym ) of anti-hermitian operators on Hsym . Proof. We will prove the corresponding statements for the complexifications: lTC,C = (µ) lTC ⊕ ilTC = B(Hsym ). The proposition then follows from taking only anti-hermitian operators on both sides. Now note that HTC,± ∈ lTC,C since we can express them as linear combinations of HTC,2 with the commutator of HTC,1 and HTC,2 . Furthermore, (µ) (µ) XM act as µ1 on Hsym . Hence, Eq. (46) shows that the restriction lTC,C is generated by (µ) 1, Y3 and HTC,± considered as operators on Hsym . Note that all operators in this proof (µ) are operators on Hsym , and therefore we simplify the notation by dropping temporarily the superscript µ, when operators are concerned. (µ) The first step is to show that Y3k , Y3j HTC,± ∈ lTC,C holds for all k, j ∈ N0 . This is done by induction. The statement is true for k ∈ {0, 1} and j = 0. Now assume it holds for all k ∈ {0, . . . , n} and j ∈ {1, . . . , n−1}. Lemma 4.13(i) shows that the commutator [Y3n−1 HTC,+ , HTC,− ] is a polynomial in Y3 with −(n + 2)Y3n+1 as leading term. Since (µ) (µ) Y3j ∈ lTC,C for j ∈ {0, . . . , n} we can subtract all lower order terms and get Y3n+1 ∈ lTC . To handle Y3n HTC,± we use Lemma 4.13(ii). The commutator [Y3n+1 , HTC,+ ] is of the (µ) form P (Y3)HTC,+ with an nth -order polynomial P . Since Y3k HTC,+ ∈ lTC,C , we can (µ) subtract all terms of order k < n and conclude that Y3n HTC,+ ∈ lTC,C . Now consider a polynomial P with P (ν) = 0 for ν 6= κ and P (κ) = 1 with ν, κ ∈ (µ) (µ) {0, . . . , dµ }. Since all Y3n are in lTC,C , we get |µ; κihµ; κ| = P (Y3 ) ∈ lTC,C . Applying the (µ) same argument to Y3n HTC,± , we also get |µ; κihµ; κ ± 1| ∈ lTC,C and the general case |µ; νihµ; λ| with µ 6= λ can be treated with repeated commutators of |µ; κihµ; κ ± 1| for different values of κ. This proposition says that the control system with Hamiltonians HTC,j with j ∈ (µ0 ) {1, 2, 3} can generate any special unitary U (µ0 ) on Hsym for any µ0 . However, some calculations using computer algebra, we have done for the case M = 2 indicate that


25

we cannot exhaust all of SU (XM ). In other words: After U (µ0 ) is fixed, we loose the (µ) possibility to choose an arbitrary U (µ) ∈ SU (Hsym ) for another µ. Our analysis for two atoms suggests that the Lie algebra generated by the HTC,j is almost as big as su(X2 ), but does not contain operators of the form A ⊗ 1 with a diagonal traceless operator A (except HTC,1 ). This observation suggests the choice of the Hamiltonians HCC,k with k ∈ {1, . . . , M+1} in Eq. (27), which lead to a dynamical group exhausting SU (XM ). This is shown in the next proposition, which completes the proof of Thm. 3.6. Proposition 4.15. The dynamical group generated by HCC,k with k ∈ {1, . . . , M+1} coincides with SU (XM ). Proof. Let us introduce the operators κ(k, j) ∈ uC (XM ) (the complexification of u(XM )) given by κ(k, j)(µ) = |µ; kihµ; j| with k, j ∈ {0, . . . , M} and κ(k, j) = 0 if k ≥ dµ and j ≤ dµ , where dµ = min(µ, M+1); cf. Eq. (44). We can re-express Y± in terms of PM −1 P κ(k, j) as Y+ = k=0 κ(k+1, k), Y− = M k=1 κ(k−1, k). Compare this to the definition of Y± in (45). The truncation of the sums occuring for µ < M is now built into the definition of the κ(k, j). Similarly we can write the HCC,j for j ∈ {1, . . . , M} as HCC,j = κ(k, k) − κ(k−1, k−1). The κ(k, j) are particularly useful because their commutator has the following simple form: [κ(k, j), κ(p, q)] = δjp κ(k, q) − δkq κ(p, j). Note that all truncations for small µ are automatically respected. This can be used to calculate the commutator of HCC,k and Y± . To this end we introduce the M × M matrix (Ajk ) with Ajj = 2, Aj,k = −1 if |j − k| = 1 and Ajk = 0 otherwise. Using (Ajk ) we can P write [HCC,j , Y+ ] = k Ajk κ(k, k−1). The matrix (Ajk ) is tridiagonal, and therefore its determinant can be easily calculated and it equals M+1. Hence (Ajk ) is invertible, and we can express κ(j, j−1) for j ∈ {1, . . . , M} as linear combination of the commutators [HCC,k , Y+ ]. Now consider the Lie algebra lCC generated by HCC,k with k ∈ {1, . . . , M+1} and its complexification lCC,C . We have HTC,1 ∈ lCC since it can be written as a linear combination of the HCC,j . In addition HTC,3 = HCC,M +1 ∈ lCC and since S+ ⊗ a, S− ⊗ a∗ can be written as (complex) linear combinations of HTC,3 and its commutator with HTC,1 we get S+ ⊗ a, S− ⊗ a∗ ∈ lCC,C . To calculate the commutators [HCC,j , S+ ⊗ a] note that according to (45) we have S+ ⊗ a = f (XM , Y3)Y+ and f (XM , Y+ ) commutes with HCC,k . Hence [HCC,j , S+ ⊗ a] = [HCC,j , f (XM , Y3 )Y+ ] = f (XM , Y3 )[HCC,j , Y+ ] = P k Ajk f (XM , Y3 )κ(k, k−1). Using the reasoning from the last paragraph, we see that f (XM , Y3)κ(k, k−1) ∈ lCC . Similarly we can show by using commutators with S− ⊗ a∗ that all κ(k, k+1)f (XM , Y3) are in lCC , too. By expanding the function f we see in this way that for k ∈ {1, ..., M} the operators A+ = P (k) κ(k, k−1), A− = P (k) κ(k−1, k), A3 = κ(k, k) − κ(k−1, k−1) (53) p with P (k) := XM + (1−k)1 are elements of lCC,C . To conclude the proof, we apply again Corollary 4.6. Hence we have to consider [K] the truncated algebra lCC . To this end, look at the subalgebra lCC,k of lCC generated by the operators in (53). They are acting on the subspace generated by basis vectors |µ; ki,


26

|µ; k−1i and if we write A1 = A+ + A− , A2 = i(A+ − A− ) we get (up to an additive shift in the operator XM ) the same structure already analyzed in Lemma 4.7 (cf. also the operators Aα,k in Eq. (37)). Hence we can apply the method from Sect. 4.2 to see that for all µ ∈ {0, . . . , K} the operators |µ; kihµ; k| − |µ, k−1ihµ, k−1|, |µ; kihµ; k−1| [K] and |µ; k−1ihµ; k| are elements of lCC,C (provided k ≤ dµ ). Now we can generate all operators |µ; pihµ, j| with p, j ≤ dµ by repeated commutators of |kihk−1| and |k−1ihk| (µ) [K] for different values of k. This shows that suC (Hsym ) ⊂ lCC,C for all µ ≤ K. By passing [K]

to anti-selfadjoint elements we conclude that lCC = su(XM )[K] holds for all K. Hence the statement follows from Corollary 4.6. 5. Strong controllability

The purpose of this section is to show how one can complement the block-diagonal dynamical groups from the last section to get strong controllability. We add one generator which breaks the abelian symmetry of the block-diagonal decomposition. The proofs for pure-state controllability and strong controllability are given in Proposition 5.2 and Proposition 5.6, respectively. This completes the proof of Theorems 3.2, 3.4 and 3.7. 5.1. Pure-state controllability Consider a family H1 , . . . , Hn of control Hamiltonians on the Hilbert space H with joint domain D ⊂ H admitting a U(1)-symmetry defined by a charge operator X with the same domain. Since all the subspaces H(µ) are invariant under all time evolutions, which can be constructed from the Hk , pure-state controllability cannot be achieved. For rectifying this problem, we have to add a Hamiltonian that breaks this symmetry in a specific way. We will do so by using complementary operators as in Definition 2.2. Hence in addition to the projections X (µ) , µ ∈ N0 we have the mutually orthogonal projections Eα , α ∈ {+, 0, −} introduced in Sect. 3 and the corresponding derived (µ) structures. This includes in particular the subprojections Xα ≤ X (µ) , µ ∈ N0 and the (µ) (µ) Hilbert spaces Hα onto which they project. Recall, that they satisfy Xα = Eα X (µ) (µ) (µ) (µ) (µ) and X (µ) = X− ⊕ X0 ⊕ X+ , and that for µ > 0 the X± are required to be non-zero. (K+1) For the following discussion we need in addition the Hilbert spaces H[K] = H[K] ⊕H− , the projections F[K] onto them and the group SU (X, F[K] ) of U ∈ SU (X) commuting with F[K] . Furthermore we will indicate restrictions to the subspaces H[K] by a subscript [K], e.g. SU [K](X, F[K] ) denotes the corresponding restriction of SU (X, F[K]) which has (K+1) the form SU [K] (X, F[K]) = SU [K](X) ⊕ SU (X− ). Now one can prove the following lemma, which will be of importance in the subsequent subsections. Lemma 5.1. Consider a strongly continuous representation π : U(1) → U(H) with charge operator X, an operator H complementary to X, and the objects just introduced. For all K ∈ N, introduce the Lie group GX,F,K generated by SU [K] (X, F[K]), exp(itH), t ∈ R and global phases exp(iα)1, α ∈ [0, 2π). Then the group GX,F,K acts transitively on the unit sphere of H[K] .


27

(µ) Proof. Consider φ ∈ H[K] and choose U˜1 ∈ SU [K] (X, F[K] ) such that X+ U˜1 φ = 0 for all µ > 0. This is possible, since SU (H(µ) ) acts transitively (up to a phase) on the unit (µ) (µ) (µ) vectors of H(µ) = H− ⊕H0 ⊕H+ . According to item (ii) of Def. 2.2 we can find t ∈ R (K+1) (K) (e.g. t = π/2 will do) such that exp(itH)H+ = H− holds. Hence exp(itH)φ ∈ H[K] ˜2 ∈ SU [K] (X, F[K] ) with φ1 = U˜2 exp(itH)U˜1 φ ∈ H[K−1] . Applying and we can find a U this procedure K times we get φK = UK · · · U1 φ ∈ H[0] with Uj ∈ GX,F,k . Similarly we can find V1 , . . . , VK ∈ GX,F,k with ψK = Vk · · · V1 ψ ∈ H[0] . Now note that the group GX,F,0 can be regarded as a subgroup of GX,F,k (which acts trivially on the orthocomplement of H[0] in H[K]). Hence, the statement of the lemma follows from the fact that, due to condition (iii) of Def. 2.2, the group GX,F,0 acts transitively on the unit vectors in K[0] = F[0] H.

The first easy consequence of this lemma is the following result which is a proof of Thm. 2.3 which we restate here as a proposition. Proposition 5.2. Consider a strongly continuous representation π : U(1) → U(H) with charge operator X and a family of selfadjoint operators H1 , . . . , Hd on H. Assume that the following conditions hold: (i) All eigenvalues µ of X are greater than or equal to 0. (ii) H1 , . . . , Hd−1 commute with X. (iii) The dynamical group generated by H1 , . . . , Hd−1 contains SU (X). (iv) The operator Hd is complementary to X.

Then the system (1) with Hamiltonians H0 = 1, H1 , . . . , Hd is pure-state controllable. Proof. We have to show that for each pair of pure states ψ, φ ∈ H and each ǫ > 0 there is a finite sequence Uk ∈ U(H) with k ∈ {1, . . . , N} and either Uk ∈ SU (X), Uk = exp(itHd ), or Uk = exp(iα)1 such that kψ − UN · · · U1 φk < ǫ. To this end, first note that we can find K ∈ N such that kψ − F[K] ψk < ǫ/3 and kφ − F[K]ψk < ǫ/3, where F[K] is the projection defined in the first paragraph of this subsection. Therefore kψ − UN · · · U1 φk ≤ kψ−F[K] ψk+kF[K]ψ−UN · · · U1 F[K]φk+kUN · · · U1 F[K] φ−UN · · · U1 φk < ǫ provided kF[K] ψ − UN · · · U1 F[K] φk < ǫ/3. Hence we can assume that ψ, φ ∈ H[K] and apply Lemma 5.1. This leads to a sequence V1 , . . . , VN ∈ GX,F,K with VN · · · V1 φ = ψ. Now note that the dynamical group G generated by H0 , . . . , Hd contains by assumption the group SU (X), the unitaries exp(itHd ) and the global phases exp(iα)1. Hence with the definition of GX,F,K , we get for j ∈ {1, . . . , N} a Wj ∈ G with [Wj , F[K]] = 0 and F[K] Wj = Vj , and therefore ψ = WN · · · W1 φ. But by definition the dynamical group is the strong closure of monomials UN · · · U1 with Uj = exp(itj Hkj ) for some tj ∈ R and kj ∈ {0, . . . , d + 1}. In other words for all U ∈ G, ξ ∈ H and ǫ > 0 we can find such a monomial satisfying kUN · · · U1 ξ − Uξk < ǫ. Applying this statement to the operators Wj and the vectors Wj−1 · · · W1 φ concludes the proof.


28

This proposition can be applied to all systems studied in Sect. 3. Therefore, they are all pure-state controllable. However, as already stated, one can even prove strong controllability, which is the next goal. 5.2. Approximating unitaries Lemma 5.1 shows that the group GX,F,K acts transitively on the pure states in the Hilbert space H[K] . This implies that there are only two possibilities for this group: either GX,F,K coincides with group of symplectic unitaries on H[K] (which is only possible if the dimension of H[K] is even), or it is the whole unitary group [46, 47, 48]. At the same time we have seen in Prop. 5.2 that (under appropriate conditions on the control Hamiltonians) each U ∈ GX,F,K admits an element W in the dynamical group satisfying W ξ = Uξ for all ξ ∈ H[K]. Proving full controllability can therefore be reduced to two steps: (i) Find arguments that for an infinite number of K ∈ N, the group GX,F,K cannot be unitary symplectic, such that it has to coincide with the full unitary group on H[K] .

(ii) Show that each unitary U ∈ U(H) can be approximated by a sequence WK , K ∈ N of unitaries of the form WK = Uk ⊕Vk , where Uk ∈ U(H[K] ) can be chosen arbitrarily, while VK is a unitary on (1 − F[K] )H which is (at least partly) fixed by the choice of Uk .

The purpose of this subsection is to prove the second statement, while the first one is postponed to Section 5.3. We start with the following lemma: Lemma 5.3. Consider a sequence F[K] , K ∈ N of finite-rank projections converging strongly to 1 and satisfying F[K] F[K+1] . For each unitary U ∈ U(H) there is a sequence U[K], K ∈ N of partial isometries, which converges strongly to U and satisfies ∗ ∗ U[K] U[K] = U[U ] U[K] = F[K] ; i.e. F[K] is the source and the target projection of U[K] . Proof. Let us start by introducing the space D ⊂ H of vectors ξ ∈ H satisfying F[K] ξ = ξ for a K ∈ N. It is a dense subset of H and we can define the map m : D → N, m(ξ) = min{K ∈ N | F[K]ξ = ξ}. All operators in this proof are elements of the unit ball B1 (H) = {A ∈ B(H) | kAk ≤ 1} in B(H). A sequence AK of elements of B1 (H) converges to A ∈ B1 (H) iff limK→∞ AK ξ = Aξ holds for all ξ ∈ D; [45, I.3.1.2]. Now define A[K] = F[K] UF[K] . For ξ ∈ D, we have UF[K] ξ = Uξ if K > m(ξ) and limK→∞ F[K]Uξ = Uξ since F[K] converges strongly to 1. Hence the strong limit of the A[K] is U, similarly one can show that the strong limit of A∗[K] is U ∗ . The A[K] are not partial isometries. We will rectify this problem by looking at the polar decomposition. To this end, first consider |A[K]|2 = A∗[K] A[K] and kA∗[K]A[K] ξ − ξk = kA∗[K] A[K] ξ − U ∗ Uξk ≤ kA∗[K] (A[K] − U)ξk + k(A∗[K] − U ∗ )Uξk ≤ k(A[K] − U)ξk + k(A∗[K] − U ∗ )Uξk where we have used that kA∗[K] k ≤ 1 holds. Strong convergence of A[K] and A∗[K] implies limK→∞ kA∗[K]A[K] ξ − ξk = 0. Hence |A[K] |2 converges strongly to 1. The operators A[K] are of finite rank with support and range contained in H[K] = F[K] H. Hence the |A[K]| have pure point spectrum and their spectral decomposition

Controlling Several Atoms in a Cavity 29 P is λ∈σ(|A[K] |) λPλ with eigenvalues 0 ≤ λ ≤ 1 and spectral projections Pλ satisfying Pλ ≤ F[K] for λ > 0. Using the fact that the Pλ are mutually orthogonal, we get P P for |A[K]|2 : k|A[K]|2 ξ − ξk = k λ∈σ(|A[K] |) (λ2 − 1)Pλ φk = λ∈σ(|A[K] |) |λ2 − 1|kPλξk = P P λ∈σ(|A[K] |) |λ − 1|(λ + 1)kPλ ξk ≥ λ∈σ(|A[K] |) |λ − 1|kPλ ξk. Hence strong convergence of |A[K]|2 implies strong convergence of |A[K]|. Now we can look at the polar decomposition A[K] = W[K]|A[K] |. The W[K] are partial isometries, and moreover, since support and range of the A[K] are contained in ∗ ∗ K[K], they satisfy W[K] W[K] ≤ F[K] and W[K] W[K] ≤ F[K]. In other words, we can look upon the W[K] as partial isometries on the finite dimensional Hilbert space H[K]. As such we can extend them to untaries U[K] ∈ U(H[K] ) without sacrificing the relation to A[K] , i.e. A[K] = U[K] |A[K]|. As operators on H, the U[K] are still partial isometries, but now with source and target projection equal to F[K] as stated in the lemma. The only remaining point is to show that the U[K] converges strongly to U. This follows from kU[K] ξ − Uξk ≤ kU[K] ξ − A[K] ξk + kA[K]ξ − Uξk and kU[K] ξ − A[K] ξk = kU[K] (1 − |A[K]|)ξk since A[K] converges strongly to U and |A[K] | to 1. Now we come back to the case discussed in the beginning of this subsection under item (ii): Lemma 5.4. Consider U, F[K] and U[K] as in Lemma 5.3, and an additional sequence ∗ ∗ of partial isometries V[K] , K ∈ N with V[K] V[K] = V[K] V[K] = 1 − F[K] . The operators W[K] = U[K] + V[K] are unitary, and if U is the strong limit of the U[K], the same is true for the W[K] . Proof. The kernels of U[K] and V[K] are (1 − F[K] )H and H[K] = F[K] H, respectively. These spaces are complementary, and therefore W[K] = U[K] + V[K] is unitary for all K. To show strong convergence, recall the space D and the function D ∋ ξ 7→ m(ξ) ∈ N introduced in the last proof. For ξ ∈ F we have W[K]ξ = U[K] ξ if K > m(ξ). Hence by assumption limK→∞ W[K] ξ = limK→∞ U[K] ξ = Uξ, which implies strong convergence of W[K] to U. 5.3. Strong controllability We are now prepared to prove Theorem 3.4. The first step is the following lemma announced already at the beginning of Subsection 5.2. Lemma 5.5. Consider the group GX,F,K introduced in Lemma 5.1 and assume that there is a µ ≤ K with d(µ) = dim(H(µ) ) > 2. Then GX,F,K = U(H[K]). Proof. Consider the group SG X,F,K consisting of elements of GX,F,K with determinant 1. By Lemma 5.1 this group acts transitively on the set of pure states of the Hilbert space H[K]. Hence, there are only two possibilities left††: SG X,F,K coincides either with the unitary symplectic group USp(H[K]) or with the full unitary group U(H[K] ); †† Note that H[K] is a finite-dimensional Hilbert space. Hence after fixing a basis e1 , . . . , ed it can be identified with Cd .


30

cf. [46, 47, 48]. Assume SG X,F,K = USp(K[K] ) holds. This would imply that SG X,F,K is self-conjugate (or more precisely the representation given by the identity map on SG X,F,K ⊂ B(H[K] ) is self-conjugate). In other words, there would be a unitary V ∈ ¯ denotes complex conjugation in U(H[K] ) with V UV ∗ = U¯ for all U ∈ SG X,F,K . Here U an arbitrary but fixed basis (cf. footnote 9). Now consider SU (H(µ) ) with d(µ) > 2. It can be identified with SU(d) in its first fundamental representation λ1 (i.e. the “defining” representation). At the same time it is a subgroup of SG X,F,K (one which acts nontrivially only on H(µ) ⊂ H[K] ). Existence of a V as in the last paragraph would imply that λ1 is unitarily equivalent to its conjugate representation, which is the d − 1st fundamental representation. This is impossible if d(µ) > 2 holds. Hence V with the described properties does not exist and SG X,F,k has to coincide with SU (H[K] ) and therefore GX,F,K = U(H[K] ) as stated. Finally we can conclude the proof of Thm. 3.4 which we restate here as the following proposition: Proposition 5.6. A control system (1) with control Hamiltonians H0 = 1, . . . , Hd satisfying the conditions from Prop. 5.2 is strongly controllable, if d(µ) = dim H(µ) > 2 for at least one µ ∈ N. Proof. Consider an arbitrary unitary U ∈ U(H). By Lemma 5.3, there is a sequence of partial isometries U[K] converging strongly to U, and by Lemma 5.5 we can assume that U[K] ∈ GX,F,K . Now considering the dynamical group G generated by the Hj , define the subgroup G(F[K]) of U ∈ G commuting with F[K] , and the restriction G[K] of G(F[K] ) to H[K] . The assumptions on the Hj imply that G[K] = GX,F,K = U(H[K] ). Hence there is a sequence WK , K ∈ N of unitaries with W[K] ∈ G(F[K] ) ⊂ G and F[K]W[K] = U[K]. Since U[K] converges to U strongly, Lemma 5.4 implies that the strong limit of the W[K] is U, which was to show. This proposition shows strong controllability for all the systems studied in Sect. 3. The only exception is one atom interacting with one harmonic oscillator (Sect. 3.1). Here we have d(µ) = dim H(µ) ≤ 2 and we can actually find a unitary V with V UV = U¯ for all U ∈ SU [K] (X1 , F[K] ). However, the elements U of SU (X1 ) are block diagonal where the blocks U (µ) ∈ SU (H(µ) ) can be chosen independently. This implies V ∈ SU K] (X1 , F[K]), which is incompatible with V HJC,4 V ∗ = −HJC,4 (cf. Eq. (15) for the definition of HJC,4 ) which would be necessary for the group GX1 ,F,K to be self-conjugate. Hence we can proceed as in the proof of Prop. 5.6 to prove Thm. 3.2. 6. Conclusions and Outlook Many of the difficulties of quantum control theory in infinite dimensions arise from the fact that, due to unbounded operators, the group U(H) of all unitaries on an infinitedimensional separable Hilbert space H is in fact no Lie group as long as it is equipped


31

Table 1. Controllability results for several 2-level atoms in a cavity as derived here.

System

Control Hamiltonians

one atom HJC,j , j = 1, 2, Eq. (9) HJC,j , j = 1, 2, Eq. (9) HJC,4 , Eq. 15 M atoms HIC,j , j = 1, . . . 2M with individual controls of Eq. (18)

———— Controllability ———— system algebra g, dynamic group G g = su(X1 ), G = SU (X1 )

[Thm. 3.1]

strongly controllablea with G = U(H)

[Thm. 3.2]

g = su(XM ) and G = SU (XM )

[Thm. 3.3]

HIC,j , j = 1, . . . 2M + 1 strongly controllablea with individual controls of Eqs. (18,22) with G = U(H) g ⊂ u(XM ) and G ⊂ U(XM )

[Thm. 3.5]

HCC,j , j = 1, . . . , M + 1 under collective control of Eq. (27)

g = su(XM ) and G = SU (XM )

[Thm. 3.6]

HCC,j , j = 1, . . . , M + 2 under collective control of Eq. (27)

strongly controllablea with G = U(H)

[Thm. 3.7]

M atoms HTC,j , j = 1, 2, 3 under collective control of Eq. (23)

a

[Thm. 3.4]

Here in the strong topology, no system algebra or exponential map exists.

with the strong topology, which inevitably is the correct choice when studying questions of quantum dynamics. Yet U(H) contains a plethora of subgroups which are still infinite-dimensional while admitting a proper Lie structure – including in particular a Lie algebra l consisting of unbounded operators and a well-defined exponential map. An important example are those unitaries with an abelian U(1)-symmetry, which in the Jaynes-Cummings model relates to a kind of particle-number operator. As shown here, this infinite-dimensional system Lie algebra l can be exploited for control theory in infinite dimensions in close analogy to the finite-dimensional case. Due to the in-born symmetry of l and the corresponding Lie group G, full controllability cannot be achieved that way. Yet we have also shown that this problem can readily be overcome by complementary methods directly on the group level. For several 2-level atoms interacting with one harmonic oscillator (e.g., a cavity mode or a phonon mode), these methods allowed us to extend previous results substantially, in particular in two aspects also summarized in Table 1: (A) We have answered approximate control and convergence questions for asymptotically vanishing control error. (B) Our results include not only reachability of states, but also its operator lift, i.e. simulability of unitary gates. To this end, we have introduced the notion of strong controllability, and we have shown that all systems under consideration require only a


32

fairly small set of control Hamiltonians for guaranteeing strong controllability, i.e. simulability. — Thus we anticipate the methods introduced here will find wide application to systematically characterize experimental set-ups of cavity QED and ion-traps in terms of pure-state controllability and simulability. Acknowledgements This work was supported in part by the eu through the integrated programmes q-essence and siqs, and the eu-strep coquit, and moreover by the Bavarian Excellence Network enb via the international doctorate programme of excellence Quantum Computing, Control, and Communication (qccc), by Deutsche Forschungsgemeinschaft (dfg) in the collaborative research centre sfb 631 as well as the international research group for 1482 through the grant schu 1374/2-1.

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